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Symmetric Cryptography CSE 484 / CSE M 584 (Spring 2012) Symmetric Cryptography Tadayoshi Kohno Thanks to Dan Boneh, Dieter Gollmann, Dan Halperin, John Manferdelli, John Mitchell, Vitaly Shmatikov, Bennet Yee, and many others for sample slides and materials ... Goals for Today Cryptography (Symmetric) Physical Security, Computer Security, and Cryptography Also • Lab due on Friday • 584 reading up, due next Tuesday • HW1 out Tues or Wed Broad Class of Hash Functions hash function H message message x. “digest” .y x’’. y’ x’. bit strings of any length n-bit bit strings H is a lossy compression function • Collisions: h(x)=h(x’) for distinct inputs x, x’ • Result of hashing should “look random” (make this precise later) – Intuition: half of digest bits are “1”; any bit in digest is “1” half the time Cryptographic hash function needs a few properties… One-Way Intuition: hash should be hard to invert • “Preimage resistance” • Let h(x’)=y∈{0,1}n for a random x’ • Given y, it should be hard to find any x such that h(x) =y How hard? • Brute-force: try every possible x, see if h(x)=y • SHA-1 (common hash function) has 160-bit output – Expect to try 2159 inputs before finding one that hashes to y. Collision Resistance Should be hard to find distinct x, x’ such that h(x)=h(x’) • Brute-force collision search is only O(2n/2), not O(2n) • For SHA-1, this means O(280) vs. O(2160) Birthday paradox (informal) • Let t be the number of values x,x’,x’’… we need to look at before finding the first pair x,x’ s.t. h(x)=h(x’) • What is probability of collision for each pair x,x’? 1/2n • How many pairs would we need to look at before finding the first collision? O(2n) • How many pairs x,x’ total? Choose(t,2)=t(t-1)/2 ∼ O(t2) • What is t? 2n/2 One-Way vs. Collision Resistance One-wayness does not imply collision resistance • Suppose g is one-way • Define h(x) as g(x’) where x’ is x except the last bit – h is one-way (to invert h, must invert g) – Collisions for h are easy to find: for any x, h(x0)=h(x1) Collision resistance does not imply one-wayness • Suppose g is collision-resistant • Define h(x) to be 0x if x is n-bit long, 1g(x) otherwise – Collisions for h are hard to find: if y starts with 0, then there are no collisions, if y starts with 1, then must find collisions in g – h is not one way: half of all y’s (those whose first bit is 0) are easy to invert (how?); random y is invertible with probab. 1/2 Weak Collision Resistance Given randomly chosen x, hard to find x’ such that h(x)=h(x’) • Attacker must find collision for a specific x. By contrast, to break collision resistance it is enough to find any collision. • Brute-force attack requires O(2n) time • AKA second-preimage collision resistance Weak collision resistance does not imply collision resistance Which Property Do We Need? UNIX passwords stored as hash(password) • Weak collision resistance: hard to recover the/a valid password Integrity of software distribution • Weak collision resistance (second-preimage resistance) • But software images are not really random… • Collision resistance if considering malicious developers Auction bidding • Alice wants to bid B, sends H(B), later reveals B • One-wayness: rival bidders should not recover B (this may mean that she needs to hash some randomness with B too) • Collision resistance: Alice should not be able to change her mind to bid B’ such that H(B)=H(B’) Common Hash Functions MD5 • 128-bit output • Designed by Ron Rivest, used very widely • Collision-resistance broken (summer of 2004) RIPEMD-160 • 160-bit variant of MD5 SHA-1 (Secure Hash Algorithm) • 160-bit output • US government (NIST) standard as of 1993-95 • Also recently broken! (Theoretically -- not practical.) SHA-256, SHA-512, SHA-224, SHA-384 SHA-3: Forthcoming. Basic Structure of SHA-1 (Not Required) Against padding attacks Split message into 512-bit blocks 160-bit buffer (5 registers) Compression function initialized with magic values • Applied to each 512-bit block and current 160-bit buffer • This is the heart of SHA-1 How Strong Is SHA-1? Every bit of output depends on every bit of input • Very important property for collision-resistance Brute-force inversion requires 2160 ops, birthday attack on collision resistance requires 280 ops Some weaknesses, e.g., collisions can be found in 263 ops (2005) International Criminal Tribunal for Rwanda (Example Application) http://www.nytimes.com/2009/01/27/science/ 27arch.html?_r=1&ref=science Credits: Alexei Czeskis, Karl Koscher, Batya Friedman HMAC Construct MAC by applying a cryptographic hash function to message and key Invented by Bellare, Canetti, and Krawczyk (1996) Mandatory for IP security, also used in SSL/TLS Structure of HMAC magic value (flips half of key bits) Secret key padded to block size Block size of embedded hash function another magic value (flips different key bits) Embedded hash function (strength of HMAC relies on strength of this hash function) “Black box”: can use this HMAC construction with any hash function (why is this important?) “Amplify” key material (get two keys out of one) Very common problem: given a small secret, how to hash(key,hash(key,message)) derive a lot of new keys? Achieving Both Privacy and Integrity Authenticated encryption scheme Recall: Often desire both privacy and integrity. (For SSH, SSL, IPsec, etc.) M C M/invalid Encrypt Decrypt Alice K K Bob K K Key . K Message . M Ciphertext . C Adversary Some subtleties! Encrypt-and-MAC Natural approach for authenticated encryption: Combine an encryption scheme and a MAC. EKe,Km DKe,Km Return M if M M valid EncryptKe MACKm DecryptKe VerifyKm valid/invalid C’ T C’ T Ciphertext Ciphertext But insecure! [BN, Kra] Assume Alice sends messages: FIREM1 DON’TM2 FIRE FIREM3 EncryptKe MACKm EncryptKe MACKm EncryptKe MACKm C’1 TT11 C’2 T2 C’3 TT33 If Ti = Tj then Mi = Mj Adversary learns whether two plaintexts are equal. Especially problematic when M1, M2, ... take on only a small number of possible values. Results of [BN00,Kra01] M M MACKm M EncryptKe M T EncryptKe MACKm C’ MACKm EncryptKe C’ T C’ T C Ciphertext C Ciphertext C Ciphertext C Encrypt-then-MAC MAC-then-Encrypt Encrypt-and-MAC Privacy Strong (CCA) Weak (CPA) Insecure Integrity Strong (CTXT) Weak (PTXT) Weak (PTXT).
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